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arXiv · 1811.11010

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

Patricia Alonso-Ruiz·Fabrice Baudoin·Li Chen·Luke Rogers·Nageswari Shanmugalingam·Alexander Teplyaev
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Abstract

We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincar\'e inequality and a weak Bakry-\'Emery curvature type condition, this BV class is identified with the heat semigroup based Besov class $\mathbf{B}^{1,1/2}(X)$ that was introduced in our previous paper. Assuming furthermore a quasi Bakry-\'Emery curvature type condition, we identify the Sobolev class $W^{1,p}(X)$ with $\mathbf{B}^{p,1/2}(X)$ for $p>1$. Consequences of those identifications in terms of isoperimetric and Sobolev inequalities with sharp exponents are given.

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Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev. 2018-11-27. Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates. https://arxiv.org/abs/1811.11010

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